Download Situation 1 - Mr Idea Hamster

Situation 1. A population has a Normally distributed variable x. From this population
we draw a Simple Random Sample of size n. The mean, µ, of the population is unknown.
The standard deviation of the population, σ, is known. The mean of the sample, x , is
computed. The population variable x has the distribution N(µ, σ) and the sample mean
x has the distribution N(, n ) Here is the picture:
Population
x distribution -> N(µ, σ)
µ = 
?
σ is known
Simple Random Sample
x distribution -> N(, n )
mean = x
size = n
Confidence Intervals

We can estimate µ with
  x  z* / n
Where z* is determined by the confidence level C. The table below gives z* for some
common values of C:

C
z*
90%
1.645
95%
1.960
99%
2.576
Hypothesis Testing.
The Null Hypothesis:
H0: µ = µ0
The Alternative Hypothesis:
Ha: µ > µ0
First compute the z statistic using the formula:
z
x  0
/ n
Then for significance level alpha determine z*. Then if z > z* we reject the null
hypothesis and say that we have statistically significant evidence for the alternative
hypothesis. For the other alternative hypotheses, Ha: µ > µ0 or Ha: µ  µ0, we adjust
the test appropriately.
Situation 2. A population has a Normally distributed variable x. From this population
we draw a Simple Random Sample of size n. The mean, µ, of the population is unknown.
The standard deviation of the population, σ, is also unknown. From the sample we
compute both the mean x and the standard deviation s. The population variable x has the
distribution N(µ, σ) and the sample mean x has the distribution t-distribution with mean
with n-1 degrees of freedom.
Here is the picture:
Population
x distribution -> N(µ, σ)
µ=?
σ=?
Simple Random Sample
x distribution -> t(µ, dof = n-1)
mean = x
size = n
Confidence Intervals
We can estimate µ with
  x  t *s/ n
Where t* is determined by the confidence level C and the degrees of freedom of the t
distribution (n-1). The table below is similar to the table for the z procedures:

C
t*
90%
95%
99%
Depends on degrees of
freedom
Hypothesis Testing.
The Null Hypothesis:
H0: µ = µ0
The Alternative Hypothesis:
Ha: µ > µ0
First compute the t statistic using the formula:
t

x  0
s/ n
Then for significance level alpha determine t*. Then if t > t* we reject the null
hypothesis and say that we have statistically significant evidence for the alternative
hypothesis. For the other alternative hypotheses, Ha: µ > µ0 or Ha: µ  µ0, we adjust
the test appropriately.
Situation 3. A proportion p of a population has some particular outcome of interest.
From this population we draw a Simple Random Sample of size n. The proportion, p, of
the population is unknown. The proportion of the sample with the outcome of interest is
number of successes in the sample
computed as p 
. For a sufficiently large sample size
n
the p statistic has the distribution N( p,
p(1 p )
n
)
is the picture:
Here

Population

proportion p “have it”
Simple Random Sample
p distribution -> N( p,
size = n
p=?
p(1 p )
n
)
Confidence Intervals
We can estimate p with
p  p  z*


p(1 p)
n
Where z* is determined by the confidence level C. The table below gives z* for some
common values of C:

C
z*
90%
1.645
95%
1.960
99%
2.576
Hypothesis Testing.
The Null Hypothesis:
H0: p = p0
The Alternative Hypothesis:
Ha: p > p0
First compute the z statistic using the formula:
z

p  p0
p o (1 p 0 )
n
Then for significance level alpha determine z*. Then if z > z* we reject the null
hypothesis and say that we have statistically significant evidence for the alternative
hypothesis. For the other alternative hypotheses, Ha: p > p0 or Ha: p  p0, we adjust
the test appropriately.